Reliability assessment of transformer insulating oil using accelerated life testing

To improve the reliability and reduce the maintenance cost of transformer oil, a life prediction of transformer oil is needed so that the maintenance of transformer can be performed correctly. However, it is difficult to predict the reliable lifetime of transformer oil accurately because of its unkind operating condition at different environments. To solve this problem, based on the theory of accelerated life testing (ALT), a reliability assessment method for transformer insulating oil on a normal operational conditions is proposed. An inverse power Weibull distribution model for insulating oil lifetime with voltage is built. Numerical procedure of model parameter estimation is presented, the variances of model parameters and reliability indices, which including mean lifetime, reliability, reliable lifetime at given reliability and failure rate, are derived. The feasibility and correctness of the proposed method are validated by real lifetime data of transformer insulating oil in literature. The reliability of transformer insulating oil used at normal usage conditions are predicted by the proposed method, and the point and interval estimations of reliability indices are evaluated. The results show that reliable lifetime and mean lifetime under reliability limit should be considered simultaneously in repair or replacement of transformer insulating oil.

www.nature.com/scientificreports/ failure times of insulation oil generally follow Weibull distribution [30][31][32] . Therefore, a Weibull IPL life model is applied in this paper to analyze the reliability and reliable life of transformer insulation oil. Therefore, the novelty and main contributions of this work can be summarized as follows: 1. Numerical method of model parameter estimation of IPL-Weibull distribution for insulating oil lifetime is proposed, the variances of model parameters and reliability indices, which including mean lifetime, reliability, reliable lifetime and failure rate, are derived. 2. The reliability of transformer insulating oil used at usual operational conditions are predicted by the proposed method, and the point and interval estimations of reliability indices are evaluated. The feasibility and correctness of the proposed method are validated by real lifetime data of transformer insulating oil in literature.
The paper is organized as follows. In "Methodology", the details on lifetime data modelling for transformer insulating oil using IPL-Weibull distribution, model parameter estimation, as well as the point and interval estimation for reliability indices are given. In "Case study", some information about ALT for a real case in literature are described. Results and comparison with the existing method are presented with details in "Results and discussion". Conclusions are drawn in the final and concluding section.

Methodology
Reliability modelling for transformer insulating oil with accelerated life testing. Some basic assumptions of ALT for insulating oil are given as follows: 1. The lifetime of insulating oil follows the Weibull distribution under usual use stress level and accelerated stress level; 2. The failure mechanism of insulating oil is the same at different stress levels; 3. The acceleration model of insulating oil is the same at each stress level.
Suppose that failure time t of insulating oil follows a two-parameter Weibull distribution, then the cumulative distribution function (CDF) F(t) and probability density function (pdf) f(t) of failure time t of insulating oil are respectively given by and where η is scale parameter and β is shape parameter, η > 0, β > 0.
The IPL model can be used for insulating oil where voltage is the main stress. Therefore, its characteristic life is where K > 0, N > 0 are model parameters, and U > 0 is voltage stress. Therefore, from Eqs. (1) and (3), the reliability R of insulating oil at given time t can be given by Numerical method of model parameter estimation. Assume that n stress levels of insulating oil in Thus, the corresponding log-likelihood function is Taking the first partial derivatives of Eq. (7) with respect to the model parameters β, K and N, and setting them equal to zero, respectively. The maximum likelihood estimates (MLE) of model parameters satisfy Eqs. (8)- (10): www.nature.com/scientificreports/ However, Eqs. (8)- (10) have no closed-form solutions, and a numerical method, which consists of three steps, is needed.
Step 1: Estimate the first model parameter β using the least square method. Twice taking the logarithmic of Eq. (1) with some mathematical transformations, the following Eq. (11) can be obtained as where F i (t i, j ) can be given by midpoint estimate or median rank estimate, and the estimators are and Therefore, the least square method 33 can be used to obtain an estimate of the parameter β i by combining Eq. (11) with Eqs. (12) or (13). ALT does not change failure mechanism, and thus the estimators of β i (i = 1, 2, …, n) should be equal each other. However, there exit some errors and uncertainty in test, the estimators of β i are not exactly the same. Therefore, its weighted estimated value can be given by  www.nature.com/scientificreports/ Step 3: Estimate the last model parameter N by the iterative method. Finally, substituting the estimated value N into Eq. (8), the estimated value K of the last parameter K can be obtain, and its (r + 1) th iteration value is where K (r) 1 is the r th iteration value of parameter K. When the estimated value K of parameter K is obtained by using Eq. (9) or Eq. (10), its value is or In general, at this case, K 1 � =K 2 =K 3 , the estimated value K can be selected according to the maximum log-likelihood value of Eq. (7), and given by Thus, all estimated values of three model parameters can be obtained.

Interval estimates for model parameters and reliability indices.
To estimate the confidence bounds of model parameters and reliability indicators, their variances are needed. The variance-covariance matrix of model parameters can be given by Fisher information matrix method as follows 34 : Commonly used reliability indices include mean time-to-failure t MTTF , the reliability R with a given time t, the reliable life t R for a given reliability and failure rate function.
Based on Eqs. (2) and (4), t MTTF can be obtained as follows: and the variance of t MTTF is where Г(·) is gamma function, and according to Eq. (4), the reliable life t R for a specified reliability can be given by According to Eqs. (2) and (4), the failure rate function of insulating oil is got by To obtain the confidence bound of reliability, reliable life and failure rate of insulating oil, the equivalent change method is applied. In Eq. (4), set u = ln KU N t β = β(ln K + N ln U + ln t) , then IE ln θ = ln θ ± z α/2 Var ln θ = ln θ ± z α/2 1 IE θ =θ exp ±z α/2 Var θ θ = θ exp −z α/2 Var θ θ ,θ exp z α/2 Var θ θ  (27) and (32), the interval estimation of reliability R can be given by The confidence interval [v L , v P ] of v can also be estimated using Eqs. (25), (27) and (34). Finally, the interval estimate of reliable life is given as Finally, in Eq. (31), set w = ln = ln β + β(ln K + N ln U) + (β − 1) ln t , then The interval estimate of failure rate can also be given as

Results and discussion
When the least square method is used to estimate shape parameter β of accelerated life model, the midpoint estimate and median rank estimate can be used to fit the reliability of insulating oil. The estimates β of shape parameter β are shown in Table 2. It can be seen that the different estimate methods give different results. At the same time, even the same method is used, the estimated results for shape parameter β in the different stress level i are also different.  It can be seen from Fig. 1 that the slope of failure probability line at two stress levels of 26 kV and 32 kV is small, and the slope of the other five voltages are basically parallel. The main reason is that there are only 3 failure data obtained at the lowest stress level of 26 kV. The first failure datum 5.79 is particularly far less than the other two data 1579.52 and 2323.70, so the estimators have a certain deviation.
Akaike information criterion (AIC) and Bayesian information criterion (BIC) 36 are most widely used in selecting the optimal model, the values of AIC and BIC are given by where p is the number of estimated parameters, q is the number of all lifetime, and maxlnL is the maximized loglikelihood. Table 3 shows the comparison results of model parameter estimation using MLE with the midpoint estimate and median rank estimate.
From Table 3, it is found that the MLE with median rank is superior to MLE with midpoint estimate, the former has a larger log-likelihood value and smaller AIC and BIC values. Compared with the result of Ref. 35 , the relative errors of log-likelihood estimation, AIC and BIC are all within 0.022%, indicating that the method proposed in this paper has a high estimation accuracy. Therefore, the estimators of model parameter are given as follows: β = 0.8017, N = 17.7318 and K = 6.70E-29, respectively.
It can be seen from Table 4, the insulating oil has a higher mean lifetime 1.4406E + 05, but its reliability at mean lifetime is very low, only 33.16%, and the reliable life of insulating oil with 90% reliability is 7689.68 min, only 5.34% of mean lifetime. At this case, if the average life is taken as the criterion of maintenance and replacement only, it will bring some potential safety hazards. Therefore, the replacement time should be determined according to the reliability and mean lifetime of the transformer during maintenance simultaneously.

Conclusions
Based on the theory of constant ALT, a probabilistic method for reliability assessment of transformer insulating oil is proposed using two-parameter Weibull IPL life model. Numerical procedure of model parameter estimation is given. The reliability indices such as the MMTF, reliability, reliable life and failure rate of insulating oil under normal service conditions were predicted by using the proposed method, and the point estimation and interval estimation were calculated. Compared with the existing method, the method proposed in this paper has a high accuracy, the relative errors of log-likelihood estimation, AIC and BIC do not exceed 0.022%. The results of reliability analyses show that even the insulating oil has a higher mean lifetime, the reliable life is not always high. Therefore, mean lifetime and reliable life should be considered simultaneously in replacement and maintenance of insulating oil.     www.nature.com/scientificreports/